#72224973

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#12030
commutator	C₂×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#346	C₂³
socle	C₂	#135826
prime radical	C₂⁴	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×C₄×M₃₂ ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,11,5,12,20,39,6,10,21,35,23,36,49,60,7,9,19,33,24,34,47,62,26,32,48,38,53,61,64,63)(3,13,15,8,16,29,18,30,17,37,44,31,22,51,46,40,14,41,42,27,43,56,45,57,25,54,55,28,52,58,50,59);(1,3)(2,8,10,28,9,27,32,31)(4,18,19,45)(5,22)(6,25)(7,14)(11,37,38,41,33,54,35,13)(12,40,36,30,34,59,61,57)(15,20,42,47)(16,23)(17,26)(21,50,48,46)(24,52)(29,60,51,39,56,63,58,62)(43,53)(44,64,55,49);(2,9)(4,19)(8,27)(10,32)(12,34)(15,42)(18,45)(20,47)(21,48)(28,31)(30,57)(36,61)(40,59)(44,55)(46,50)(49,64)

Polycyclic presentation:

2^10|a^2=f,b^2=d,d^2=g,f^2=h,h^2=i,i^2=j,b^a=bfhij,c^a=ce,d^a=dij,e^a=ej,e^b=ej,f^b=fh,f^c=fj,f^d=fj,g^a=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224981, #72225073, #72225084
maximal normal subgroup of	#72224981, #72225073, #72225084

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);